The sum of perpetuities method (SPM)
is a way of valuing a business assuming that investors discount the future
earnings Earnings are the net benefits of a corporation's operation. Earnings is also the amount on which corporate tax is due. For an analysis of specific aspects of corporate operations several more specific terms are used as EBIT (earnings before interes ...
of a firm regardless of whether earnings are paid as
dividends
A dividend is a distribution of profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-in ...
or retained. SPM is an alternative to the
Gordon growth model In finance and investing, the dividend discount model (DDM) is a method of valuing the price of a company's stock based on the fact that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. In ...
(GGM)
and can be applied to business or
stock valuation
In financial markets, stock valuation is the method of calculating theoretical values of companies and their stocks. The main use of these methods is to predict future market prices, or more generally, potential market prices, and thus to profit fr ...
if the business is assumed to have constant earnings and/or dividend growth. The variables are:
*
is the value of the stock or business
*
is a company's earnings
*
is the company's constant growth rate
*
is the company's risk adjusted
discount rate
*
is the company's dividend payment
::
Comparison with other models
SPM and the Walter model
SPM is a generalized version of the Walter model.
The primary difference between SPM and the Walter model is the substitution of earnings and growth in the equation. Consequently, any variable which may influence a company's constant growth rate such as inflation, external financing, and changing industry dynamics can be considered using SPM in addition to growth caused by the reinvestment of
retained earnings
The retained earnings (also known as plowback) of a corporation is the accumulated net income of the corporation that is retained by the corporation at a particular point of time, such as at the end of the reporting period. At the end of that peri ...
internally. Because
is substituted in the equation, SPM is also directly comparable to other constant growth models.
SPM and the Gordon growth model
In a special case when a company's
return on equity
The return on equity (ROE) is a measure of the profitability of a business in relation to the equity. Because shareholder's equity can be calculated by taking all assets and subtracting all liabilities, ROE can also be thought of as a return on '' ...
is equal to its risk adjusted discount rate, SPM is equivalent to the Gordon growth model (GGM). However, because GGM only considers the
present value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
of dividend payments, GGM cannot be used to value a business which does not pay dividends. Also, when a firm's return on equity is not equal to the discount rate, GGM becomes highly sensitive to input value changes. Alternatively, SPM values dividends and retained earnings separately, taking into consideration the present value of the future income generated by retained earnings, and then summing this result with the present value of expected dividends held constant in perpetuity. Consequently, SPM can be used to value a growing company regardless of
dividend policy. SPM is also much less sensitive to input value changes when a company's return on equity is different from the discount rate. An empirical test
shows that SPM is substantially more accurate in estimating observed stock market prices than the Gordon Growth Model.
SPM and the PEG ratio
The
PEG ratio The 'PEG ratio' ( price/earnings to growth ratio) is a valuation metric for determining the relative trade-off between the price of a stock, the earnings generated per share ( EPS), and the company's expected growth.
In general, the P/E ratio is h ...
is a special case in the SPM equation. If a company does not pay dividends, and its risk adjusted discount rate is equal to 10%, SPM reduces to the PEG ratio:
::
::
::
SPM can be used to help explain the PEG ratio as it provides a derivation and theoretical framework for the PEG.
Derivation of SPM
SPM is derived from the compound interest formula via the present value of a perpetuity equation. The derivation requires the additional variables
and
, where
is a company's retained earnings, and
is a company's rate of return on equity. The following relationships are used in the derivation:
::I:
::II:
Derivation
Given by relationship II, a company with perpetual life which pays all of its earnings out as dividends has a growth rate of zero. It can therefore be valued using the present value of a perpetuity equation:
::
However a company may elect to retain a portion of its earnings to produce incremental earnings and/or dividend growth. If the value of both dividends and retained earnings are considered, and the return on equity is equal to the firm's discount rate, the company could be valued by the same function (refer to relationship I):
::
Yet retained earnings are different from dividends paid, because dividend payments represent a cash ''inflow'' to a company's owners (shareholders) while retained earnings which are reinvested to produce growth are effectively an invested cash ''outflow''. Therefore, when the rate of return on equity is not equal to the discount rate, the ''present value of the future income generated by the retained earnings''
must be considered rather than the amount of earnings retained today.
::
Where
is the present value of the future income generated by the assets purchased using
. The income generated by
depends on the firm's rate of return on equity and therefore
is a function of
where
, is equal to the income produced by the assets purchased using
. Assuming perpetual life and a constant rate of return on equity,
can also be determined using the present value of a perpetuity equation:
::
Substituting
for
in the equation above produces the Walter model:
::
And given by relationship II,
is equal to
. Substituting the term,
into the above equation produces the SPM constant growth valuation model:
::
Limitations of SPM
The SPM equation requires that all variables be held constant over time which may be unreasonable in many cases. These include the assumption of constant earnings and/or dividend growth, an unchanging dividend policy, and a constant risk profile for the firm. Outside financing may not be considered unless the financing is perpetually recurring as
capital structure
In corporate finance, capital structure refers to the mix of various forms of external funds, known as capital, used to finance a business. It consists of shareholders' equity, debt (borrowed funds), and preferred stock, and is detailed in the ...
must also be held constant.
References
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Valuation (finance)